Discretization methods for semilinear parabolic optimal control problems

Chryssoverghi, I

dc.contributor.author	Chryssoverghi, I	en
dc.date.accessioned	2014-03-01T01:24:01Z
dc.date.available	2014-03-01T01:24:01Z
dc.date.issued	2006	en
dc.identifier.issn	1705-5105	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/17164
dc.relation.uri	http://www.math.ualberta.ca/ijnam/Volume-3-2006/No-4-06/2006-04-04.pdf	en
dc.subject	optimal control	en
dc.subject	parabolic systems	en
dc.subject	discretization	en
dc.subject	piecewise bilinear controls	en
dc.subject	penalized gradient projection method	en
dc.subject	relaxed controls	en
dc.subject.classification	Mathematics, Applied	en
dc.subject.classification	Mathematics	en
dc.subject.other	NONCONVEX VARIATIONAL-PROBLEMS	en
dc.subject.other	APPROXIMATION	en
dc.title	Discretization methods for semilinear parabolic optimal control problems	en
heal.type	journalArticle	en
heal.language	English	en
heal.publicationDate	2006	en
heal.abstract	We consider an optimal control problem described by semilinear parabolic partial differential equations, with control and state constraints. Since this problem may have no classical solutions, it is also formulated in the relaxed form. The classical control problem is then discretized by using a finite element method in space and the implicit Crank-Nicolson midpoint scheme in time, while the controls are approximated by classical controls that are bilinear on pairs of blocks. We prove that strong accumulation points in L-2 of sequences of optimal (resp. admissible and extremal) discrete controls are optimal (resp. admissible and weakly extremal classical) for the continuous classical problem, and that relaxed accumulation points of sequences of optimal (resp. admissible and extremal relaxed) discrete controls are optimal (resp. admissible and weakly extremal relaxed) for the continuous relaxed problem. We then apply a penalized gradient projection method to each discrete problem, and also a progressively refining version of the discrete method to the continuous classical problem. Under appropriate assumptions, we prove that accumulation points of sequences generated by the first method are admissible and extremal for the discrete problem, and that strong classical (resp. relaxed) accumulation points of sequences of discrete controls generated by the second method are admissible and weakly extremal classical (resp. relaxed) for the continuous classical (resp. relaxed) problem. For nonconvex problems whose solutions are non-classical, we show that we can apply the above methods to the problem formulated in Camkrelidze relaxed form. Finally, numerical examples are given.	en
heal.publisher	ISCI-INSTITUTE SCIENTIFIC COMPUTING & INFORMATION	en
heal.journalName	INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING	en
dc.identifier.isi	ISI:000241069500004	en
dc.identifier.volume	3	en
dc.identifier.issue	4	en
dc.identifier.spage	437	en
dc.identifier.epage	458	en